Stellar Black Holes Can "Stretch" Supermassive Black Hole Accretion Disks

Star formation in the outer regions of the accretion disks of active galactic nuclei (AGNs) is a widely studied topic (e.g., Shlosman & Begelman 1989; Collin & Zahn 1999; Goodman & Tan 2004; Levin 2007; Wang et al. 2011, 2023b; Cantiello et al. 2021; Chen et al. 2023 ). The outer regions of the accretion disk are far from the central supermassive black hole (SMBH), where self-gravity plays a dominant role, and the disk inevitably collapses to form stars (e.g., Kolykhalov & Syunyaev 1980; Shlosman & Begelman 1989). Nuclear star formation can also be triggered by jetted tidal disruption events in SMBHs (e.g., Perna & Grishin 2022). In addition, stars near the center of the galaxy can interact with the SMBH accretion disk, resulting in a loss of orbital energy and angular momentum, and are captured by the accretion disk (e.g., Syer et al. 1991; Cantiello et al. 2021). The redshift-independent high metal abundance in quasar broad-line regions provides strong observational evidence for star formation activities in SMBH accretion disks (e.g., Artymowicz et al. 1993; Hamann & Ferland 1999; Wang et al. 2011, 2022; Qi et al. 2022). Stars in accretion disks can accrete gas and rapidly evolve into compact objects, such as white dwarfs, neutron stars, and stellar black holes (sBHs), as first pointed out by Cheng & Wang (1999).

The discovery of the Fermi bubbles strongly suggests that Sagittarius A* was an active AGN in the past (Su et al. 2010; H. Y. K. Yang et al. 2022). During this active stage, stars or compact objects may form in the accretion disk of Sagittarius A*. Although the SMBH accretion in Sagittarius A* is now quenched, the present-day stellar and compact object populations should be affected by star formation epochs during the past active phase of Sagittarius A*. Very recently, Gravity Collaboration et al. (2023) found that the Sagittarius A* flare is located at about 9 gravitational radii, consistent with the Keplerian orbital motion of the innermost accretion zone hot spot. While the physical origin of the flare remains unclear, it is proposed that a star (Leibowitz 2020) or a ∼40 M_⊙ sBH (Wang et al. 2023a) orbiting periodically around the central SMBH drives this Sagittarius A* flare through interaction with the SMBH accretion disk. The star or the sBH is embedded in the advection-dominated accretion flow around the SMBH in the Galactic center.

In the special environment of the AGN accretion disk, stars and compact objects are prone to events such as massive stellar explosions and mergers or collisions of compact objects. These events can produce transient emission such as core-collapse supernovae (e.g., Grishin et al. 2021), γ-ray bursts (e.g., Cheng & Wang 1999; Ray et al. 2023), neutrino bursts (e.g., Zhu et al. 2021), tidal disruption events (e.g., Y. Yang et al. 2022), Bondi explosions (e.g., Wang et al. 2021), and binary black hole mergers (e.g., Graham et al. 2020). Mergers of binary black holes, black hole–neutron star systems, or binary neutron stars are important sources of gravitational waves detected by LIGO/Virgo/KAGRA (Abbott et al. 2019). GW190521 (Abbott et al. 2020) is of considerable interest because the Zwicky Transient Facility detected its potential electromagnetic counterpart in AGN J124942.3+344929 (Graham et al. 2020). This implies that the binary black hole merger may occur in the AGN accretion disk (but see Palmese et al. 2021 for the discussion of an association by chance), which has been used to infer cosmological parameters (e.g., Chen et al. 2022).

Although there have been many studies of sBHs in AGN accretion disks, we remain poorly informed about their number densities and distributions. Stars (e.g., Thompson et al. 2005) or sBHs (e.g., Gilbaum & Stone 2022) in the self-gravity regions of AGN accretion disks can substantially increase the central temperature of gas. Here we show that sBHs embedded in the static standard disk (hereafter SSD; Shakura & Sunyaev 1973) modify the effective temperature distribution of the accretion disk and change the disk half-light radii at various wavelengths. Different sBH distributions in the accretion disk have different degrees of influence on the half-light radius, which may be used to infer sBH distributions.

Quasar microlensing observations provide a unique opportunity to resolve AGN accretion disks. Distant quasars might be gravitationally lensed by foreground galaxies and have multiple lensed images. There are time delays in the flux variations of different lensed images that can be used to constrain the cosmological model (i.e., the strong-lensing time-delay cosmography; for a review, see, e.g., Treu & Marshall 2016). On top of the strong-lensing effects, stellar objects (in the foreground galaxy) that might happen to be in the line of sight of a lensed quasar image can cause additional flux magnifications. These microlensing-induced apparent flux variations are size dependent and can be used to effectively measure the half-light radii of AGN accretion disks (Mortonson et al. 2005). A number of studies have measured the half-light radii of AGN accretion disks using the microlensing effects; the results suggest that the measured half-light radii are 2–4 times larger than the SSD predictions (e.g., Morgan et al. 2010).

In this work, we propose that quasar microlensing observations can be used to probe sBHs in AGN accretion disks. The Letter is organized as follows. In Section 2, we detail the model calculations; in Section 3, we present the spectral energy distribution (SED) of the model and the half-light radius as a function of wavelength; in Section 4, the implications for the probe of the sBH distribution, the AGN accretion physics, and strong-lensing time-delay cosmography are discussed. The main conclusions are summarized in Section 5.

A large number of sBHs can form in situ or be captured from nuclear star clusters. For instance, Artymowicz et al. (1993) and Rozyczka et al. (1995) point out that the AGN accretion disk captures stars from nuclear star clusters at a rate of 10⁻⁴–10⁻³ yr⁻¹. The typical AGN lifetime is 10⁷–10⁸ yr. Thus, the total captured number of stars, which can further accrete gas and evolve into sBHs, is 10³–10⁵ within the AGN lifetime. A detailed calculation of Gilbaum & Stone (2022) by considering in situ formation, capture, and migration yields that the number of sBHs within a disk radius of 3–3000 R_s over the AGN lifetime can be 10²–10⁸ (Figure 12 in their Section 3.2). Thus, we establish a model with 10³–10⁵ (e.g., Artymowicz et al. 1993; Rozyczka et al. 1995) sBHs embedded in the AGN accretion disk. Neutron stars may outnumber sBHs in the AGN accretion disk, but their energy outputs are smaller than sBHs; white dwarfs are difficult to form within the AGN lifetime because the typical evolutionary time of a white dwarf is ∼1 billion yr (e.g., Catalán et al. 2008). Therefore, we only consider the effects of sBHs. We also assume that sBHs do not significantly affect the angular momentum transport in the AGN accretion disk (Gilbaum & Stone 2022). The SMBH mass is significantly larger than the total mass of the sBHs, and the sBHs only play an essential role for the gas in their vicinity. A small fraction of the gas (f_sBHs) in the AGN accretion disk is accreted by the sBHs rather than the central SMBH. This causes the effective temperature distribution of the AGN accretion disk to be different from the canonical T_eff ∝ R^−3/4 of the SSD theory and the contributions of the luminosities at different radii to the total luminosity to be altered for a given wavelength.

When embedded in the AGN accretion disk, sBHs can rapidly accrete gas from the AGN accretion disk. The total accretion rate for the embedded sBHs is ${f}_{\mathrm{sBHs}}{\dot{M}}_{\mathrm{tot}}$, where ${\dot{M}}_{\mathrm{tot}}$ is the total mass accretion rate in the AGN accretion disk. Each accreting sBH can produce intensive emission, preferably in the X-ray band. Given that the AGN disk is optically thick, these X-ray photons cannot escape freely but are absorbed by the AGN disk and heat the ambient gas. In this case, the AGN disk has two sources of heating: the local viscous heating in the AGN disk and the sBHs. We discuss in Section 4.2 the possible consequence if sBHs are distributed in optically thin accretion-disk regions (∼10⁴ R_S; Thompson et al. 2005). We do not consider the possible interaction between the SMBH and sBH accretion processes. We argue this may be a good approximation if f_sBHs ≪ 1.

The heating rates can be estimated as follows. For the local viscous heating in the AGN disk, the heating rate per unit area is

$$\begin{eqnarray}&&{Q}_{\mathrm{vis}}^{+}=\displaystyle \frac{3{{GM}}_{\bullet }{\dot{M}}_{\bullet }}{8\pi {R}^{3}}{f}_{r},\end{eqnarray} \tag{ 1 }$$

where G, M_•, and R are the gravitational constant, the SMBH mass, and the radius of the AGN accretion disk, respectively; the factor ${f}_{r}=1-{(3{R}_{{\rm{S}}}/R)}^{1/2}$, where R_S = 2GM_•/c² is the Schwarzschild radius of the SMBH and the mass accretion rate to the SMBH ${\dot{M}}_{\bullet }=(1-{f}_{\mathrm{sBHs}}){\dot{M}}_{\mathrm{tot}}$. Meanwhile, the total heating rate due to sBHs per unit area depends upon their distribution on the AGN accretion disk. We divide the accretion disk from the inner boundary R_in to the outer boundary R_out into 256 equal rings on a logarithmic scale. For simplification, it is straightforward to assume that sBHs are equally densely distributed in each ring; the sBH number on each ring N(R) is

$$\begin{eqnarray}&&N(R)=2\pi R{\rm{\Delta }}R\displaystyle \frac{{N}_{\mathrm{sBHs}}}{\pi ({R}_{\mathrm{out}}^{2}-{R}_{\mathrm{in}}^{2})},\end{eqnarray} \tag{ 2 }$$

where N_sBHs is the total number of sBHs in the AGN accretion disk. As a result, most accreting sBHs reside in the outer regions of the AGN accretion disk. Note that real sBH distributions can be more complicated than this, as we discuss in Section 4.1. Each ring is further divided into 256 equal zones along the azimuthal direction, i.e., Δϕ = 2π/256. Hence, the size of each zone is comparable to the scale height of the AGN accretion disk. The sBHs in each zone will then be able to heat the gas within the zone isotropically. The heating rate in each zone is simply

$$\begin{eqnarray}&&{Q}_{\mathrm{sBHs}}^{+}=N(R,\phi ){L}_{\mathrm{sBH}},\end{eqnarray} \tag{ 3 }$$

where N(R, ϕ) and L_sBH are the number of sBHs in a zone and the bolometric luminosity of each accreting sBH. We assume that all sBHs accrete at the Eddington limit, ⁷ and L_sBH equals the Eddington luminosity of an sBH, which is L_sBH,Edd = 1.26 × 10³⁸ M_sBH/M_⊙ (erg s⁻¹), where M_sBH and M_⊙ are the sBH mass and the solar mass, respectively. Here, the radiative efficiency of 10% is assumed. We stress that the radiative efficiency and the additional heating due to sBHs increase with the black hole spin. Note that the Bondi accretion rate of a typical sBH in the AGN accretion disk is much larger than the Eddington accretion rate (e.g., Wang et al. 2021). The real accretion process is more complex than the ideal Bondi case, and accompanying outflows may significantly modulate the accretion rate (e.g., Takeo et al. 2020). Hence, following Gilbaum & Stone (2022), we assume that sBHs accrete at the Eddington limit. Then, the total number of sBHs is

$$\begin{eqnarray}&&{N}_{\mathrm{sBHs}}=\displaystyle \frac{{f}_{\mathrm{sBHs}}{\dot{M}}_{\mathrm{tot}}}{{\dot{M}}_{\mathrm{sBH},\mathrm{Edd}}},\end{eqnarray} \tag{ 4 }$$

where ${\dot{M}}_{\mathrm{sBH},\mathrm{Edd}}$ is the Eddington accretion rate of an sBH, i.e.,${\dot{M}}_{\mathrm{sBH},\mathrm{Edd}}=10{L}_{\mathrm{sBH},\mathrm{Edd}}/{c}^{2}$.

The heating due to sBHs plays an important role in the outer disk zones. Indeed, it is evident that the ratio

$$\begin{eqnarray}&&\epsilon ={Q}_{\mathrm{sBHs}}^{+}/({Q}_{\mathrm{vis}}^{+}{\rm{\Delta }}S(R,\phi ))\propto {\left(R/{R}_{{\rm{S}}}\right)}^{3},\end{eqnarray} \tag{ 5 }$$

where ΔS(R, ϕ) = RΔϕΔR is the area of each zone. For the case of sBHs embedded in an AGN accretion disk, the effective temperature profile of the AGN disk can be obtained by considering the balance between the heating rate and the disk surface cooling rate. We consider perfect blackbody radiation at each radius. Hence, the effective temperature T_eff,sBHs is

$$\begin{eqnarray}&&2\sigma {T}_{\mathrm{eff},\mathrm{sBHs}}^{4}(R,\phi )={Q}_{\mathrm{vis}}^{+}(1+\epsilon ),\end{eqnarray} \tag{ 6 }$$

where σ is the Stefan–Boltzmann constant. We set R_in = 3 R_S and R_out = 3000 R_S. The effective temperature profile for the AGN disk embedded with sBHs is shown in the top right panel of Figure 1. For the sake of comparison, we also obtain the temperature profile for a pure SSD (the top left panel of Figure 1) by eliminating ${Q}_{\mathrm{sBHs}}^{+}$ in Equation (6) and fixing f_sBHs = 0. We compare the effective temperature profiles for both cases in the bottom panel of Figure 1. In the presence of sBHs, the AGN disk temperature is significantly hotter than the SSD in the outer regions. Hence, we expect that, for a given wavelength, the disk size of an AGN disk with sBHs is larger than that of a pure SSD.

Zoom In Zoom Out Reset image size

Microlensing observations can constrain the AGN disk sizes. Microlensing observations essentially measure the half-light radius (Mortonson et al. 2005). The half-light radius can be estimated as follows. We consider perfect blackbody radiation at each radius and a face-on viewing angle. The local monochromatic luminosity can be expressed as

$$\begin{eqnarray}&&{{dL}}_{\nu ,\mathrm{sBHs}}=\pi {B}_{\nu }({T}_{\mathrm{eff},\mathrm{sBHs}}){Rd}\phi {dR},\end{eqnarray} \tag{ 7 }$$

where ${B}_{\nu }({T}_{\mathrm{eff},\mathrm{sBHs}})=2h{\nu }^{3}/({c}^{2}{e}^{h\nu /{{kT}}_{\mathrm{eff},\mathrm{sBHs}}}-{c}^{2})$ is the Plank function, and h, ν, and k are the Planck constant, frequency, and the Boltzmann constant, respectively. The half-light radius R_half,sBHs at wavelength λ( = c/ν) satisfies

$$\begin{eqnarray}&&{\int }_{{R}_{\mathrm{in}}}^{{R}_{\mathrm{half},\mathrm{sBHs}}}{\int }_{0}^{2\pi }{{dL}}_{\nu ,\mathrm{sBHs}}=\displaystyle \frac{1}{2}{\int }_{{R}_{\mathrm{in}}}^{{R}_{\mathrm{out}}}{\int }_{0}^{2\pi }{{dL}}_{\nu ,\mathrm{sBHs}}.\end{eqnarray} \tag{ 8 }$$

The half-light radius of a pure SSD (R_half,SSD) can be calculated following the same methodology. The SED is ${L}_{\nu ,\mathrm{sBHs}}\,={\int }_{{R}_{\mathrm{in}}}^{{R}_{\mathrm{out}}}{\int }_{0}^{2\pi }{{dL}}_{\nu ,\mathrm{sBHs}}$.

We now calculate the SEDs (Section 3.1) and half-light radii (Section 3.2) of a pure SSD and an SSD with sBHs. Model parameters are set as follows. We consider three SMBH masses, 10⁷, 10⁸, or 10⁹ M_⊙. The mass of each sBH M_sBH is fixed at 50 M_⊙, but our results remain unchanged when other M_sBH values are assumed. The dimensionless accretion rate ${\dot{m}}_{\mathrm{tot}}={\dot{M}}_{\mathrm{tot}}/{\dot{M}}_{\bullet ,\mathrm{Edd}}$ is 0.3 (Kollmeier et al. 2006), where ${\dot{M}}_{\bullet ,\mathrm{Edd}}$ is the Eddington accretion rate of SMBH. The values of f_sBHs are taken as 0.06, 0.08, 0.10, and 0.12, respectively. Based on these parameters, the sBH number calculated from Equation (4) and is about 10³–10⁵, consistent with previous studies (e.g., Artymowicz et al. 1993; Rozyczka et al. 1995; Gilbaum & Stone 2022). The ratio of the total mass of sBHs to the SMBH mass is ${f}_{\mathrm{sBHs}}{\dot{m}}_{\mathrm{tot}}$, which is significantly less than 1. Gas in the AGN disk is still dominated by the SMBH gravity or the self-gravity. We calculate emission at wavelengths from 1000 to 8000 Å.

3.1. Spectral Energy Distribution

Previous studies have shown that star formation in the AGN accretion disk can revise the AGN SEDs (e.g., Goodman & Tan 2004; Thompson et al. 2005; Wang et al. 2023b), and the same is expected for accreting sBHs. The top panel of Figure 2 shows the SEDs of a pure SSD and an SSD embedded with sBHs for M_• = 10⁸ M_⊙ with f_sBHs = 0.1 (for a pure SSD, f_sBHs ≡ 0). At wavelengths larger than 4700 Å, the emission from the SSD embedded with sBHs is significantly larger than that from a pure SSD. This is because the main contribution of the sBHs is in the outer (long-wavelength emission) regions of the AGN accretion disk. At wavelengths shorter than 3600 Å, the monochromatic luminosity from the SSD embedded with sBHs is slightly weaker than that from a pure SSD. This is simply because a fraction of gas is accreted to sBHs rather than the SMBH. The monochromatic luminosity from the accreting sBHs is equal to that from the AGN disk at a wavelength of ∼3600 Å. For a more massive SMBH (e.g., 10⁹ M_⊙; the bottom panel of Figure 2), accreting sBHs have weak effects at these wavelengths. This is because ${T}_{\mathrm{eff},\mathrm{sBHs}}\propto {M}_{\bullet }^{-1/4}{\dot{m}}_{\mathrm{tot}}^{1/4}$ according to Equations (1) and (6), i.e., T_eff,sBHs decreases with increasing M_• for fixing ${\dot{m}}_{\mathrm{tot}}$. As a result, the contribution due to accreting sBHs is prominent at wavelengths longer than 8000 Å.

Zoom In Zoom Out Reset image size

We compare the composite spectra from Vanden Berk et al. (2001) and Selsing et al. (2016) with the SED of an SSD embedded with sBHs. We normalize the composite spectra to the SED of an SSD embedded with sBHs at 1450 Å. The composite spectrum from Vanden Berk et al. (2001) was compiled from a homogeneous sample of 2200 spectra with a median redshift of 1.25. Most spectra have i-band absolute magnitude fainter than −24 mag. Meanwhile, the composite spectrum of Selsing et al. (2016) was constructed from seven bright quasars with VLT/X-SHOOTER observations. Hence, we compare the composite spectrum of Vanden Berk et al. (2001) with the model SED for M_• = 10⁸ M_⊙ with f_sBHs = 0.1 (the top panel of Figure 2). In previous studies, the significant spectral slope change around 4700 Å is often attributed to the emission from hot dust (Vanden Berk et al. 2001), host galaxy contamination (Selsing et al. 2016), or the diffuse emission from the broad-line region clouds (e.g., Chelouche et al. 2019). Interestingly, just like the composite spectrum of Vanden Berk et al. (2001), the SED of the SSD embedded with sBHs shows an evident shallow shape for wavelengths longer than 4700 Å. The spectral slope of the SSD embedded with sBHs is almost identical to that of Vanden Berk et al. (2001) at wavelengths larger than 4700 Å. As mentioned above, this is because the main contribution of sBHs is in the outer regions of the AGN accretion disk. Our results suggest that this spectral slope variation may also be influenced by accreting sBHs in the AGN accretion disk (see also Sirko & Goodman 2003). The composite spectrum of Selsing et al. (2016) compiled from bright AGNs is compared with the model SED of a M_• = 10⁹ M_⊙ with f_sBHs = 0.1 (the bottom panel of Figure 2). In this case, accreting sBHs have weak effects on the model SED. Hence, the SED of an SSD embedded with sBHs does not show spectral shape change around 4700 Å, just like the SED of a pure SSD and that of Selsing et al. (2016). Note that our conclusions remain unchanged if we use other f_sBHs. In summary, our model provides a new possible origin of the spectral shape variation around 4700 Å for less luminous AGNs.

3.2. Half-light Radius

Accreting sBHs modify the effective temperature distribution of the AGN accretion disk (Figure 1), which can be distinguished by microlensing observations. The half-light radii at different wavelengths for an SSD embedded with sBHs can be calculated by Equation (8). Figure 3 shows the half-light radii at different wavelengths with M_• = 10⁸ M_⊙. For a given wavelength, the half-light radius increases as f_sBHs increases. The reason is that the larger f_sBHs is, the greater the contributions of the sBHs to the luminosity at large radii, resulting in a larger half-light radius. For the rest-frame wavelengths ≲3600 Å, the accreting sBHs have almost no influence on the half-light radius. This is because the main contribution of sBHs is at the outer regions of the AGN accretion disk, whose effective temperatures are too cold to produce short-wavelength emission. If more sBHs are distributed in the short-wavelength radiation regions, the short-wavelength half-light radii will increase (see Section 4.1). For rest-frame wavelengths larger than 5600 Å, an SSD embedded with sBHs can yield half-light radii larger than 3 times that of a pure SSD, consistent with microlensing observations.

Zoom In Zoom Out Reset image size

The range of wavelengths affected by accreting sBHs widens as the SMBH mass decreases (e.g., 10⁷ M_⊙; Figure 4). This is because the effective temperature of an SSD embedded with sBHs ${T}_{\mathrm{eff},\mathrm{sBHs}}\propto {M}_{\bullet }^{-1/4}{\dot{m}}_{\mathrm{tot}}^{1/4}{(R/{R}_{{\rm{S}}})}^{-3/4}{(1+\epsilon )}^{1/4}$ according to Equations (1) and (6). For the same R/R_S and ${\dot{m}}_{\mathrm{tot}}$, T_eff,sBHs increases with decreasing M_•. As a result, the contribution of accreting sBHs in outer regions can lead to short-wavelength emission increases with a decrease in the SMBH mass. With our hypothetical sBH distribution, the accretion disk embedded with sBHs of an SMBH with M_• = 10⁷ M_⊙ has half-light radii 3 times or more of a pure SSD for wavelengths longer than ∼3000 Å.

Zoom In Zoom Out Reset image size

The relationship between the half-light radius and wavelength is significantly altered by accreting sBHs in the AGN accretion disk. The Legacy Survey of Space and Time (LSST; Ivezić et al. 2019) and the Wide Field Survey Telescope (T. Wang et al. 2023) will perform high-resolution multiband surveys in the south and north skies, respectively. They can measure the half-light radii of some lensed quasars in multiple bands, which can verify our model calculations and probe the distributions of accreting sBHs in AGN accretion disks.

4.1. Stellar Black Hole Distributions

Accreting sBHs have significant effects on the SEDs and half-light radii of long-wavelength optical emission. Although we fix R_out at 3000 R_S, it is possible that some sBHs are located at >3000 R_S and can also have a significant impact on the SEDs and half-light radii at infrared bands. Meanwhile, similar to planets in stellar disks, sBHs can interact with gas in the AGN accretion disk and migrate inward or outward. Detailed calculations suggest that the disk can form several migration traps at which migrating sBHs are trapped and accumulated (e.g., Bellovary et al. 2016; Grishin et al. 2023). Grishin et al. (2023) note that the migration traps are approximately at 10³–10⁵ gravitational radii. Thus, we consider a case where all sBHs are trapped at the same radius of an AGN accretion disk. Figure 5 shows the results for all sBHs distributed at a radius of 1000 R_S for an AGN disk with M_BH = 10⁸ M_⊙. As expected, sBHs now only affect the effective temperature at 1000 R_S. The corresponding half-light radii are larger than a pure SSD even on wavelengths as short as 2000 Å. This is because when all sBHs are at 1000 R_S, they are closer to the radiation region of the short wavelengths than the sBH distribution of Equation (2). As a result, these sBHs have a larger impact on the half-light radii of short-wavelength emission than the sBH distribution of Equation (2).

Zoom In Zoom Out Reset image size

Real sBH distributions, which are likely to be much more complex than we have explored, can have various impacts on the half-light radii and the effects can be understood as follows. Consider a general sBH distribution of $N{(R)\propto (R/{R}_{{\rm{S}}})}^{\beta }$ with β = 2 identical to the sBH distribution of Equation (2). The ratio of the sBH heating rate to the local viscous heating rate (Equation (5)) is $\epsilon \propto {(R/{R}_{{\rm{S}}})}^{\beta +1}$. According to Equations (1) and (6), the corresponding effective temperature ${T}_{\mathrm{eff},\mathrm{sBHs}}\propto {M}_{\bullet }^{-1/4}{\dot{m}}_{\mathrm{tot}}^{1/4}{(R/{R}_{{\rm{S}}})}^{-3/4}{(1+\epsilon )}^{1/4}$. As β decreases (i.e., more sBHs are distributed in the inner regions), there are two consequences. First, the critical radius (R_c) at which = 1 moves inward. The additional heating due to sBHs only plays an important role in the disk regions outside R_c. As R_c moves inward, the shorter-wavelength (∼λ_c ≡ hc/(kT_eff(R_c))) emission regions are also affected by sBHs. As a result, the half-light radii at short wavelengths (≃λ_c) can increase. Second, the absolute value of is smaller in outer regions. As a result, the half-light radii at long wavelengths (≫λ_c) decrease. Hence, the tendency of the half-light radius with wavelength is a prospective method for inferring the sBH distribution. Inferring the sBH distribution requires long rest-frame wavelength half-light radius measurements. RXS J113155.4-123155 is one of the nearest gravitationally lensed quasars found to date, with a redshift of 0.658 (Sluse et al. 2003) and M_• = 10^7.78 M_⊙ (Peng et al. 2006). The arrows in Figures 3, 4, and 5 represent the rest-frame wavelengths of this source probed by LSST filters. The LSST is able to obtain the half-light radii in the rizy bands that are significantly influenced by sBHs.

Collisions and mergers of sBHs in an AGN accretion disk are a possible source of gravitational waves. For an AGN with M_• = 10⁸ M_⊙ and f_sBHs = 0.1, the maximum number of sBHs in the zones at 3000 R_S is 24 for the sBH distribution of Equation (2). Whitehead et al. (2023) show that two sBHs will form a binary when their radial distance is [1.85, 2.4]r_H, where r_H is the Hill radius, which is comparable to the zone size we considered in Section 2. The results of Whitehead et al. (2023) are obtained for an AGN accretion disk with M_• = 4 × 10⁶ M_⊙. The merger may become more difficult for a more massive SMBH accretion disk as the AGN disk gas density is lower. Nevertheless, we speculate that sBHs in each zone of the accretion disk can merge to form a massive sBH. We stress that our results for the SEDs and half-light radii are independent of the masses of sBHs.

4.2. AGN Accretion Physics

4.3. Implications for Strong-lensing Time-delay Cosmography

We have explored the SED and half-light radius of an AGN accretion disk embedded with accreting sBHs. We have shown that a population of accreting sBHs residing in the outer regions of an AGN accretion disk can produce observable features in SEDs and microlensing observations. The main conclusions can be summarized as follows.

• 1.  
  Embedded sBHs can cause the effective temperature of the outer regions to be significantly higher than that of the pure SSD (see Section 2; Figure 1).
• 2.  
  Compared to a pure SSD, an SSD embedded with sBHs produces a redder SED, which may contribute significantly to the spectral slope change around 5000 Å for AGNs with low (≲10⁸ M_⊙) SMBH masses (see Section 3.1, Figure 2).
• 3.  
  The dependence of the half-light radius with wavelength for an SSD embedded with sBHs significantly differs from that of a pure SSD. With suitable sBH distributions, the model half-light radius is consistent with microlensing observations (see Sections 3.2 and 4.1; Figures 3, 4, and 5).
• 4.  
  The dependence of the half-light radius with wavelength can probe the sBH distribution in the AGN accretion disk (see Sections 3.2 and 4.1; Figures 3 and 5).

As mentioned in Section 4.2, the primary influence of sBHs is at the outer regions of the AGN accretion disk; the inner regions may be more dominated by other mechanisms, such as the inhomogeneous disk proposed by Dexter & Agol (2011). Therefore, we will consider the accretion-disk size in conjunction with sBHs and other models in future works.

We thank the referee for constructive comments that improved the manuscript. We acknowledge support from the National Key R&D Program of China (No. 2023YFA1607903, No. 2023YFA1608100). S.Y.Z. and M.Y.S. acknowledge support from the National Natural Science Foundation of China (NSFC-12322303), and the Natural Science Foundation of Fujian Province of China (No. 2022J06002). J.M.W. acknowledges support from the National Natural Science Foundation of China (NSFC-11991050, NSFC-12333003). J.X.W. acknowledges support from the National Natural Science Foundation of China (NSFC-12033006, NSFC-12192221). Y.Q.X. acknowledges support from the National Natural Science Foundation of China (NSFC-12025303, NSFC-12393814) and the Strategic Priority Research Program of the Chinese Academy of Sciences (grant No. XDB0550300).

Software: Matplotlib (Hunter 2007), Numpy (Harris et al. 2020), Scipy (Virtanen et al. 2020).